Elastic net frequency-difference generalized inverse beamforming based on an iterative regularization matrix

To overcome the frequency limitations inherent in uniform array designs and to achieve effective spatial aliasing suppression and resolution enhancement for out-of-band high-frequency signals, this paper proposes an elastic net frequency-difference generalized inverse beamforming method based on an iterative regularization matrix. To address the inherent spatial resolution deficiency of conventional frequency-difference beamforming, the frequency-difference concept is introduced into the generalized inverse beamforming (GIB), leading to the development of frequency-difference generalized inverse beamforming (FD-GIB). The FD-GIB method suppresses spatial aliasing for out-of-band high-frequency signals and achieves a modest improvement in resolution; however, its resolution still falls short of practical requirements for noise source localization and identification. Therefore, to further exploit the spatial sparsity of the source distribution while ensuring solution robustness, a joint l 1 and l 2 norm constraint is introduced into FD-GIB, giving rise to the elastic net frequency-difference generalized inverse beamforming (Net-FD-GIB). This approach is formulated as a Lasso-type problem and efficiently solved using the Fast Iterative Shrinkage Thresholding Algorithm (FISTA). Although Net-FD-GIB yields improved resolution compared to FD-GIB, further enhancement remains achievable. To this end, an iterative regularization matrix replaces the identity matrix in Net-FD-GIB, thereby strengthening the penalty applied to non-source regions. This modification leads to the iterative regularization matrix-based elastic net frequency-difference generalized inverse beamforming (IRM-Net-FD-GIB), which is also solved as a Lasso-like problem via FISTA. Both simulation and experimental results demonstrate that the proposed IRM-Net-FD-GIB method effectively overcomes the frequency limitations imposed by the array design bandwidth. It achieves substantial suppression of spatial aliasing and significantly enhances resolution, while maintaining robust performance under low signal-to-noise ratio ( SNR ) conditions. • Frequency-difference generalized inverse beamforming (FD-GIB) for suppressing spatial aliasing of out-of-band high-frequency signals. • Formulates Net-FD-GIB and IRM-Net-FD-GIB as Lasso-type problems and solves them via FISTA. • IRM-Net-FD-GIB suppresses spatial aliasing of out-of-band high-frequency signals while enhancing spatial resolution. • IRM-Net-FD-GIB maintains robust performance under low SNR conditions.